Question

For a longitudinal sound wave, ($B$ is the engineering modulus of the medium and $\rho$ is the density):

• A. $\frac{d^{2}y}{d{x}^2}=\frac{B}{\rho }\frac{d^{2}y}{d{t}^2}$
• B. $\frac{dy}{dx}=\frac{B}{\rho }\frac{dy}{dt}$
• C. $\frac{d^{2}y}{d{t}^2}=\frac{B}{\rho }\frac{d^{2}y}{d{x}^2}$
• D. $\frac{dy}{dt}=\frac{B}{\rho }\frac{dy}{dx}$

Answer 1

The correct option is C $\frac{d^{2}y}{d{t}^2}=\frac{B}{\rho }\frac{d^{2}y}{d{x}^2}$

Differential equation of longitudinal sound wave $\frac{d^{2}y}{d{t}^2}=v^2\frac{d^{2}y}{d{x}^2}$
where $v$ is the velocity of sound wave.
Velocity of sound in a medium is given by $v=\sqrt{\frac{B}{\rho }}$
$\therefore$ $\frac{d^{2}y}{d{t}^2}=\frac{B}{\rho }\frac{d^{2}y}{d{x}^2}$
$⟹$ $\frac{d^{2}y}{d{x}^2}=\frac{\rho }{B}\frac{d^{2}y}{d{t}^2}$
Correct answer is option C.